Global Harmonic Analysis

PI应证明关于经典力学和量子力学之间关系的严格结果。薛定谔在1927年提出了量子力学，以解决电子如何同时运动和静止的难题。他的答案是，具有一定能量的量子粒子只能存在于稳态态。稳态态是指支配物理学的薛定谔算符的本征函数。问题是，本征函数很难可视化和理解。全局调和分析是关于在这些本征函数和物理系统背后的经典力学之间建立严格的关系。提案中的项目解决了两种相反的问题：(I)本征函数可以有多大，以及它大的点是如何分布的？这些都是最有可能找到粒子的点。(Ii)特征函数为零的点是如何分布的？这些点是它最不可能找到粒子的地方。全局调和分析是利用黎曼流形的测地线流的长时间动力学来理解特征函数的高频渐近性。其中最重要的两个问题是(1)分析特征函数的Lp范数和(2)分析节点集和临界点集。对于较大的p，Sogge和PI证明了只有当存在一个自焦点x时，才能得到普适的Lp的界。在最近的工作中，PI证明了如果度量是实解析的，那么所有的测地线都回环，并且第一个返回映射在单位切线空间上保持在单位切向空间上的有限测度。在两维空间中，所有通过x的测地线都是光滑闭合的。一个项目是将最后一个结果推广到更高的维度和平滑的指标。研究低Lp范数也是非常有趣的，我们正在利用Kakeya-Nikodym极大函数来做这件事。关于节点集和临界点集，J.Jung和PI最近证明了具有非空凹边界的非正曲面上Neumann或Dirichlet特征函数的节点域的个数必然与特征值趋于无穷大。研究还表明，特征函数有多个临界点。在未来的工作中，PI将证明一个更定量的结果，并研究更高维的情况。PI还在实际解析情况下使用特征函数的全纯延拓，以获得对节点集和临界点集的更多控制。

The PI shall prove rigorous results about the relations between classical and quantum mechanics. Schrodinger introduced quantum mechanics in 1927 to solve the puzzle of how an electron can be moving and stationary at the same time. His answer was that a quantum particle with a definite energy can only exist in a stationary state, the physics word for an eigenfunction of the Schrodinger operator governing the physics. The problem is that eigenfunctions are very hard to visualize and understand. Global Harmonic Analysis is about making rigorous relations between these eigenfunctions and the classical mechanics underlying the physical system. The projects in the proposal address two opposite kinds of questions: (i) how large can the eigenfunction be, and how are the points where it is large distributed? These are the points where it is most probable to find the particle. (ii) How are the points where the eigenfunction is zero distributed? These are the points where it is least likely to find the particle. Global Harmonic Analysis is the use of the long-time dynamics of the geodesic flow of a Riemannian manifold to understand the high-frequency asymptotics of eigenfunctions. Two of the most important problems are (1) to analyze Lp norms of eigenfunctions and (2) to analyze nodal sets and critical point sets. For large p, Sogge and the PI have shown that the universal Lp bounds are only obtained if there exists a self-focal point x where a positive measure of geodesics from x loop back to x. In recent work, The PI showed that if the metric is real analytic, then all geodesics loop back and the first return map preserves a finite measure on the unit tangent space at x in the class of Lebesgue measure. In two dimensions, it follows that all geodesics through x are smoothly closed. One project is to generalize the last result to higher dimensions and smooth metrics. It is also very interesting to study low Lp norms and we are working with the Kakeya-Nikodym maximal function to do that. Regarding nodal and critical point sets, J. Jung and the PI have recently shown that the number of nodal domains of Neumann or Dirichlet eigenfunctions on non-positively curved surfaces with non-empty concave boundary must tend to infinity with the eigenvalue. It was also showed that the eigenfunctions have many critical points. In future work, the PI shall prove a more quantitative result and to study the higher dimensional case. The PI is also using holomorphic extensions of eigenfunction in the real analytic case to gain more control over nodal and critical point sets.